Interferometric spectroscopy is founded on the relationship between the optical phase delay .DELTA..phi. which a beam of light undergoes on traversing a distance s: ##EQU1## where .lambda. is the wavelength of the light, .sigma. is the wavenumber, and s is the effective optical pathlength, taking into account the refractive characteristics of the media through which the light is propagated. The spectrum of wavelength components comprising a particular beam of light may yield information regarding the source of the light or the absorption or spectrochemical characteristics of any medium intervening between the light source and a spectrometer. The phase delay is subject to measurement, at least in a relative sense, and thus the spectral content may be derived by measuring .DELTA..phi. at a plurality of distances s.
In one realization of an interferometric spectrometer (a Michelson interferometer), discussed here for purposes of example only, the light beam is split and caused to traverse two nearly identical paths, differing only by a slight pathlength difference .DELTA.s that is varied as a periodic function of time. The two beams are then recombined, and detected by a square-law detector sensitive to the product of the electric field amplitudes characterizing the two light beams. The product of the field amplitudes contains constant terms as well as a cross term proportional to the sinusoid of the relative phase delay in the light beam traversing the two paths. The detector signal J is thus a function of the pathlength difference .DELTA.s which is being varied. The measurement result of detector signal as a function of pathlength difference J(.DELTA.s) is referred to as an interferogram. The spectral content B(.sigma.) may be derived from J(.DELTA.s) since the relation between s and .sigma. is known from Eqn. 1. For a polychromatic field illuminating the interferometer, the interferogram, J(.DELTA.s), and optical spectrum are a cosine Fourier transform pair, ##EQU2## and the interferogram is formally equivalent to the autocorrelation of the input optical field through the Wiener-Khitnitche theorem. Thus, an equivalent statement of the object of Fourier transform spectroscopy is the application of an interferometer to measure the optical field correlation for recovering the optical power spectrum by application of a transform.
The common practice in optical interferometry is to sample the interferogram at equispaced intervals of path difference. This is often accomplished by scanning path difference .DELTA.s as a linear function of time and sampling J(.DELTA.s) at intervals of constant period, however, scans weighted in time but constant in path difference are known in the art. The resulting equally-sampled interferogram is then transformed, using standard discrete Fourier transform (DFT) techniques, producing a spectrum, B(.sigma.), the spectral resolution of which is constant across the bandwidth of the derived spectrum. Standard spectroscopic practice is discussed, for example, in Ferraro & Basile (eds.), Fourier Transform Infrared Spectroscopy: Applications to Chemical Systems, (Academic Press, 1978), which is incorporated herein by reference.
For a number of reasons, however, uniform sampling is not always optimal. For example, the object of the spectroscopy might be the detection of a particular spectral feature, such that sampling to confirm or exclude the presence of that feature is of greater significance than the absolute calibration of spectral amplitudes with respect to a baseline. In other cases, data acquisition, transmission, or storage might be at a premium, and a reduction in the number of sampled points required to achieve a specified spectral resolution might be desirable, but unattainable using known techniques.
Various methods are known for obtaining a transform of a signal sampled irregularly or randomly. Examples may be found in Marvasti, "Nonuniform Sampling," in Marks (ed.), Advanced Topics in Shannon Sampling and Interpolation Theory (Springer-Verlag, 1993), Dowla, MEM Spectral Analysis for Nonuniformly Sampled Signals, (MIT, 1981), and Kay, Modern Spectral Estimation: Theory and Application (Prentice-Hall, 1988), Korenberg, et al., "Raman Spectral Estimation via Fast Orthogonal Search," in The Analyst, vol. 122, (September, 1997), pp. 879-82, and Korenberg, Chapter 7 in Non-linear Vision: Determination of Neutral Receptive Fields, Function, and Networks, Pinter & Nabet (eds.), (CRC Press, 1992), all of which are incorporated herein by reference. These methods may obtain frequency resolutions comparable to that obtained with a DFT but with a reduced number of sample points.
Another known method for obtaining a transform of a signal sampled irregularly or randomly applies a fast orthogonal search, as described, in connection with time-series analysis, in Korenberg, "A Robust Orthogonal Algorithm for System Identification and Time-Series Analysis," Biological Cybernetics, vol. 60, pp. 267-76 (1989), and in Korenberg & Paarmann, "Orthogonal Approaches to Time-Series Analysis and System Identification,", IEEE SP Magazine, (July, 199), pp. 29-43, which are incorporated herein by reference. As discussed in detail in Korenberg & Paarmann, FOS is capable of much liner frequency resolution than a conventional Fourier series analysis, and, moreover, model order is automatically determined and no polynomial equation is required to be solved in performing the transform.
None of the methods discussed, however, provides a method for specifying the sampling intervals for optimizing particular features in the transform domain.
If the objective is particularized information with respect to R(.tau.), the autocorrelation of a function u(t), for example, spectral characteristics, it is possible, in accordance with the teachings of the invention, to choose (at least in a statistical sense) the sampling intervals on t over which u(t) is measured so as to optimize the measurement for deriving the requisite properties of R(.tau.).
Similarly, the desired characteristics to be ascertained may correspond to higher-order moments.
Spectroscopic methods are known for reducing the time required to obtain a sample resulting in a spectrum, in the transform domain, of specified resolution. For example, spatial mutliplexing may be applied, wherein a Fourier Transform spectrometer is fabricated from a fixed two-beam interferometer and an array of photosensors with equi-spaced detector elements for recording an interferogram spatially distributed across the detector array. For the same signal-to-noise, the acquisition time may be reduced by a factor (the multi-channel advantage) substantially equal to the number of detector elements. The spectral resolution, in this case, is limited by the spatial separation of the detector elements.